On the Strong Martin Conjecture
We study the following conjecture. Conjecture. Let $T$ be an $\omega$-stable theory with continuum many countable models. Then either i) $T$ has continuum many complete extensions in $L_1(T)$, or ii) some complete extension of $T$ in $L_1$ has continuum many $L_1$-types without parameters. By Shelah's proof of Vaught's conjecture for $\omega$-stable theories, we know that there are seven types of $\omega$-stable theory with continuum many countable models. We show that the conjecture is true for all but one of these seven cases. In the last case we show the existence of continuum many $L_2$-types.
Permanent link to this document: http://projecteuclid.org/euclid.jsl/1183743734
Digital Object Identifier: doi:10.2178/jsl/1183743734
Mathematical Reviews number (MathSciNet): MR1129150
Zentralblatt MATH identifier: 0743.03022